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Theorem scott0f 34302
Description: A version of scott0 8912 with non-free variables instead of distinct variables. (Contributed by Giovanni Mascellani, 19-Aug-2018.)
Hypotheses
Ref Expression
scott0f.1 𝑦𝐴
scott0f.2 𝑥𝐴
Assertion
Ref Expression
scott0f (𝐴 = ∅ ↔ {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = ∅)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦)

Proof of Theorem scott0f
Dummy variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 scott0 8912 . 2 (𝐴 = ∅ ↔ {𝑤𝐴 ∣ ∀𝑧𝐴 (rank‘𝑤) ⊆ (rank‘𝑧)} = ∅)
2 scott0f.1 . . . . . 6 𝑦𝐴
3 nfcv 2912 . . . . . 6 𝑧𝐴
4 nfv 1994 . . . . . 6 𝑧(rank‘𝑥) ⊆ (rank‘𝑦)
5 nfv 1994 . . . . . 6 𝑦(rank‘𝑥) ⊆ (rank‘𝑧)
6 fveq2 6332 . . . . . . 7 (𝑦 = 𝑧 → (rank‘𝑦) = (rank‘𝑧))
76sseq2d 3780 . . . . . 6 (𝑦 = 𝑧 → ((rank‘𝑥) ⊆ (rank‘𝑦) ↔ (rank‘𝑥) ⊆ (rank‘𝑧)))
82, 3, 4, 5, 7cbvralf 3313 . . . . 5 (∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦) ↔ ∀𝑧𝐴 (rank‘𝑥) ⊆ (rank‘𝑧))
98rabbii 3334 . . . 4 {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = {𝑥𝐴 ∣ ∀𝑧𝐴 (rank‘𝑥) ⊆ (rank‘𝑧)}
10 nfcv 2912 . . . . 5 𝑤𝐴
11 scott0f.2 . . . . 5 𝑥𝐴
12 nfv 1994 . . . . . 6 𝑥(rank‘𝑤) ⊆ (rank‘𝑧)
1311, 12nfral 3093 . . . . 5 𝑥𝑧𝐴 (rank‘𝑤) ⊆ (rank‘𝑧)
14 nfv 1994 . . . . 5 𝑤𝑧𝐴 (rank‘𝑥) ⊆ (rank‘𝑧)
15 fveq2 6332 . . . . . . 7 (𝑤 = 𝑥 → (rank‘𝑤) = (rank‘𝑥))
1615sseq1d 3779 . . . . . 6 (𝑤 = 𝑥 → ((rank‘𝑤) ⊆ (rank‘𝑧) ↔ (rank‘𝑥) ⊆ (rank‘𝑧)))
1716ralbidv 3134 . . . . 5 (𝑤 = 𝑥 → (∀𝑧𝐴 (rank‘𝑤) ⊆ (rank‘𝑧) ↔ ∀𝑧𝐴 (rank‘𝑥) ⊆ (rank‘𝑧)))
1810, 11, 13, 14, 17cbvrab 3347 . . . 4 {𝑤𝐴 ∣ ∀𝑧𝐴 (rank‘𝑤) ⊆ (rank‘𝑧)} = {𝑥𝐴 ∣ ∀𝑧𝐴 (rank‘𝑥) ⊆ (rank‘𝑧)}
199, 18eqtr4i 2795 . . 3 {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = {𝑤𝐴 ∣ ∀𝑧𝐴 (rank‘𝑤) ⊆ (rank‘𝑧)}
2019eqeq1i 2775 . 2 ({𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = ∅ ↔ {𝑤𝐴 ∣ ∀𝑧𝐴 (rank‘𝑤) ⊆ (rank‘𝑧)} = ∅)
211, 20bitr4i 267 1 (𝐴 = ∅ ↔ {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1630  wnfc 2899  wral 3060  {crab 3064  wss 3721  c0 4061  cfv 6031  rankcrnk 8789
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-int 4610  df-iun 4654  df-iin 4655  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-om 7212  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-r1 8790  df-rank 8791
This theorem is referenced by:  scottn0f  34303
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